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Output (port)- Signal is sent out (to TV)
Input- Signal goes into the device
You can't output a signal from an input, You need an HDMI splitter or a switch
Back in the day, before TVs had multiple inputs, if you wanted to connect a DVD player and watch regular TV, you would have to use a switch to choose the signal source (TV or DVD) that was fed to the TV. When technology improved the DVD would have an input for the TVs signal and an output to the TV and the switch was built into the DVD. It's likely that the cable box has switching citcuitry built in but not used just as some cable boxes have USB and ethernet connectors as well for future use as well